When a three-dimensional image constituted by piling up tomograms obtained by an X-ray CT apparatus, an MRI apparatus, etc; is projected on a projection plane, this invention relates to an interpolation method which obtains other pixel values on the same projection line by interpolation calculation by using a plurality of pixel values on the projection line.
The three-dimensional image obtained by piling up a plurality of tomograms by utilizing a computer has drawn an increasing attention as the image for observing and diagnosing an outer structure of an object and an internal cubic structure of the object. Here, the term "three-dimensional image" is not directed to display three-dimensionally the images but intends to mean that two-dimensional tomograms are piled up into a three-dimensional shape and that when the images so piled up are displayed on a two-dimensional CRT, shading is effected for example, in such a fashion that those pixels which are remote from a viewpoint are displayed darker and those which are close to the viewpoint are displayed brighter and when they overlap with one another, pixels in the foreground are displayed preferentially. Therefore, though the term should be correctly called "pseudo-three-dimensional image", it is generally called the "three-dimensional image".
The tomograms that constitute the three-dimensional image are the images obtained by an X-ray CT apparatus or an MRI (which are called the "original images"), for example. If the pitch (distance) between the adjacent tomograms is great, the image is not easy to observe. Therefore, a method of determining the pixels between the adjacent tomograms by interpolation has been proposed. An example described in "MEDICAL IMAGING TECHNOLOGY" by Yasuo Sudo et al, August 1988, page 275, is shown in FIG. 1 of the accompanying drawings.
FIG. 1 shown the example where mutually parallel two original images (CT images) 1 and 2 exist adjacent to each other, and are projected from a viewpoint E to a projection plane P. The viewpoint E is an arbitrary point, and the position of the projection plane P, too, exists at an arbitrary position. The positions of the viewpoint E and the projection plane P are determined in accordance with the object of projection and the object of display.
A variety of methods exist as the projection method from the viewpoint E to the projection plane P. Therefore, various known methods may be employed such as the one that grasps the viewpoint E as a plane and effects parallel projection by parallel projection lines from this plane to the projection plane P, or another which grasps the viewpoint E as a point and effects central projection by assuming central projection lines expanding radially from this point towards the projection plane.
Referring to FIG. 1, projection from the viewpoint E to a projected point p on the projection plane P will be described. A straight line L from the viewpoint E to the projected point p is the projection line, and each pixel on this projection line L is the pixel that must be projected to the projected point p. Here, the projection line is a line that does not vertically cross two mutually parallel original images 1 and 2. In other words, the projection line L is assumed to extend obliquely with respect to the original images 1 and 2. Symbol Q.sub.1 represents the point of intersection of the projection line L with the CT image 1 on the projection line L and Q.sub.2 represents the point of intersection with the CT image 2. The pixel value of the point Q.sub.1 corresponds to the pixel value of the CT image 1 at the corresponding position and the pixel value of the point Q.sub.2 corresponds to that of the CT image 2 at the corresponding position. When the projection line L between these points Q.sub.1 and Q.sub.2 is divided by N (N: integer), the pixel value of each point on this projection line L must be determined by interpolation.
To effect interpolation, the coordinates of each point on the projection line L connecting the point Q.sub.1 to the point Q.sub.2 must be stipulated. It will be hereby assumed that the position (x.sub.1, y.sub.1, z.sub.1) of an interpolation point A (new tomogram point) as an arbitrary point is stipulated in the x-, y- and z-coordinates system. Here, symbols x, y and z represent a spatial coordinates system for stipulating the coordinates of each point under the state where the CT images 1, 2, etc., are piled up, y represents a coordinates axis orthogonally crossing the CT images 1 and 2 that are disposed parallel to each other, and x and z represent the coordinates axes orthogonally crossing y. The pile-up position (which is also referred to as the "slice position") of each CT image can be expressed by y, and the position of the pixel at each point of each CT image can be expressed by x and z. It will be assumed that the y position of the CT image 1 is Y.sub.01 and the y position of the CT image 2 is Y.sub.02.
Now, a straight line m parallel to the y axis and passing through a point A is given. The points of intersection a.sub.5 and b.sub.5 of this line m with the CT images 1 and 2 are a.sub.5 (x.sub.1, Y.sub.01, Z.sub.1) and b.sub.5 (x.sub.1, Y.sub.02, Z.sub.1), respectively. The distance between the point of intersection a.sub.5 and the interpolation point A is (y.sub.01 -y.sub.1) and the distance between the interpolation point A and the point of intersection b.sub.5 is (Y.sub.1 -Y.sub.02). Assuming that the pixel values at the points a.sub.5 and b.sub.5 are I.sub.01 and I.sub.02, respectively, the pixel value I.sub.a at the interpolation point A can be obtained from them in accordance with the following linear interpolation formula: ##EQU1##
On the other hand, the pixel positions a.sub.5 and b.sub.5 of the CT images 1 and 2 do not practically exist in many cases because the CT pixel positions of the CT images 1 and 2 are coarser than the projection resolution. In FIG. 1, the pixel positions of the CT images 1 and 2 that are practically adjacent to one another are coarse such as a.sub.1, a.sub.2, a.sub.3, a.sub.4, . . . , b.sub.1, b.sub.2, b.sub.3, b.sub.4, etc. Therefore, the pixel values of the pixel positions a.sub.5 and b.sub.5, too, are determined by the practical neighboring pixel values by interpolation. To obtain the pixel values I.sub.01 and I.sub.02 of the pixel positions a.sub.5 and b.sub.5, for instance, there is a method which calculates the pixel values by the weighted mean of the eight neighboring points a.sub.1, a.sub.2, a.sub.3, a.sub.4, b.sub.1, b.sub.2, b.sub.3 and b.sub.4 or another which calculates them by a linear interpolation method from the neighboring two points.
The explanation given above deals with the interpolation point A. To determine the pixel value I.sub.b of another interpolation point B (x.sub.1 ', y.sub.1 ', z.sub.1 '), the points of intersection a.sub.5 ' and b.sub.5 ' with the y-axis parallel line m' are similarly obtained, the pixel values I.sub.01 ' and I.sub.02 ' of the points of intersection a.sub.5 ' and b.sub.5 ' are determined by interpolation, and linear interpolation such as equation (1) is effected between the distances (y.sub.01 -y.sub.1 ') and (y.sub.1 '-y.sub.02) from the pixel values I.sub.01 ' and I.sub.02 ' obtained by interpolation described above and the points of intersection a.sub.5 ' and b.sub.5 ', so as to obtain the pixel value I.sub.01 ' and I.sub.02 ' obtained by interpolation described above and the points of intersection a.sub.5 ' and b.sub.5 ', so as to obtain the pixel value I.sub.b of the interpolation point B.
The calculation described above is made for each basic pitch between the points Q.sub.1 and Q.sub.2 of the projection line L and the interpolation pixel value for each basic pitch is determined. Because the projection line is variously determined by its projection method, such a process is effected for all the projection lines for the CT images 1 and 2. Further, a similar process is effected to other CT images adjacent to the CT images 1 and 2 to obtain the pixel values. When the pixel values so obtained are displayed, the display content becomes extremely unclear if they are as such displayed. Therefore, a display method (called a "voxel method" or a "depth method") which displays the pixel values away from the visual point darker (to a small value) and the pixel values closer to the visual point brighter is employed. This method is referred to as a "shading process". Furthermore, image processing (plane shading processing) is carried out by removing the depth side and displaying only the pixel values on the most foreground side.
In FIG. 1, if the interpolation point A between the points Q.sub.1 and Q.sub.2 is the pixel position not having the pixel value of the CT image, the interpolation pixel values I.sub.01 and I.sub.02 at the interpolation points a.sub.5 and b.sub.5 are determined, and the pixel value I.sub.a of the interpolation point A is determined from the interpolation pixels I.sub.01 and I.sub.02 by interpolation. A similar process is necessary for the interpolation point B. In other words, an interpolation process must be executed twice for each interpolation point. More than two interpolation points exist in most cases between the points Q.sub.1 and Q.sub.2, and the interpolation process must be carried out twice each time. Further, the number of projection lines is great, too, and a similar process is executed between the adjacent CT images. As a result, the number of times the interpolation process is performed becomes enormous as a whole.
FIG. 2 shows an example of non-linear interpolation according to the prior art.
Transverse lines represent the original tomograms comprising actual measurement data and longitudinal lines represent the interpolation image obtained by interpolation. Pixels are assumed to exist at the points of intersection between the transverse lines and the longitudinal lines. To determine the pixel value of the interpolation point M' on the projection line L, a straight line ml passing through the point M' and perpendicularly crossing the original tomograms is considered, and the points of intersection A0, A1, A2 and A3 between the line ml and the original tomograms are determined by interpolation. The pixel value of the point M' is then determined by non-linear interpolation by using the pixel values of the points A0 to A3.
When the pixel value of another interpolation point M" is obtained, a straight line m2 passing through the point M" and perpendicularly crossing the original tomograms is considered in the same way as in the case of M', and the points of intersection B0, B1, B2 and B3 are determined. The pixel value of the point M" is determined by non-linear interpolation by using the pixel values of these points B0 to B3.
In other words, the pixel values of four new points of intersection must be obtained by interpolation for each interpolation point, and a long time is therefore necessary for the computation.